I think it's mostly to do with how you spell the names in different countries, as the Russian-origin names are written in cyrillics. Therefore it varies quite a bit in different contexts how the names are spelled.

Yes. The ISU uses a sort of international spelling, which is not necessarily the way the skater's name is spelled in their native alphabet. Cyrillic, in particular, gets mangled so much in ISU transliteration; for example, Plyushchenko becomes "Plushenko". I suppose that is because it is considered easier to remember for those who are not Cyrillic users. The letter which used to be transliterated as "shch" is sometimes written as "sch" in English now, as well.

A lot of names ending in "iy" get transliterated with just "i" or just "y". In the actual Cyrillic lettering, there really are two letters making the sound, i.e., "iy". Nowadays, some seem to feel that the "y" looks too much like the Cyrillic letter that stands for "u", so they hesitate to use the "y" in transliteration of the "iy" name ending. I hope I am not being too confusing. I am very tired from losing sleep, but I really did want to help explain the apparent changeability of names that were originally written in Cyrillic.

Well I have an other question?
Why Yuko should have changed her name? What was the problam with Kawaguchi? I know that now she is russian citizen, but I can't understand the decision of the fedaration...

Well I have an other question?
Why Yuko should have changed her name? What was the problam with Kawaguchi? I know that now she is Russian citizen, but I can't understand the decision of the fedaration...

Apparently someone had commented on it, the reason being that they cannot write Kawaguchi in Russian and closest name is Kawaguti. May be same reason as Yeshoua became Jesus.
iI think Kavaguti's documents being in Russian, so they didn't want to create any confusion with the name.

may be someone with understanding of russian can clarify on the subject.

For Yuko's Russian passport her name was transliterated from Japanese to Russian according to the official rules, which are a bit weird (-ti not -chi), but are recognized by both foreign offices. The result was a Cyrillic name, which 'sounds' Kavaguti.

Since she had to give up her Japanese citizenship she was also issued a Russian travel/'international' passport, which has the name written both in Cyrillic and in Latin, the former being the one in Russian in her Russian national passport and the other is transliterated from it according to some official rules, which gave Kavaguti.

And all ISU paperwork is most likely done using whatever name she 'officially' holds, which is Юко Кавагути / Yuko Kavaguti now.

Well, I'm from Hungary and that's why I asked the quiestion. As you know Maxim Zavozin is now Hungarian and he didn't have to change his name, although we write it this way: Makszim. Here there is a law: if someone can confirm that he or she by nationality isn't Hungarian but is Hungarian by citizenship he or she can write his or her name on the native lanuguage.

The main issue, however, is alphabet. All documents, except for the travel passport and international driving license, are issues in Cyrillic, therefore she had to somehow write 川口 悠子 in Russian. And there are rules for it. They might be puzzling for Japanese (I've asked around and no one can explain why anyone has ever thought that it should be -ti (-ти) and not -chi (-чи)), but they are official.

In the former times there were also pretty strict rules about transliterating Russian names into Latin alphabet (which produced a lot of confusion because it was essentially based on French), but they have switched to something like English transcription recently and I was told that in the recent years one can ask for the certain spelling (to match previously issued documents), but I doubt that Yuko would have been able to 'recover' -chi at the end, since her Russian passport clearly says -ti.